If V = {0, 1, 2}. Describe all functions T: V --> V for which T(V) = V. There are six of them. Indicate which ones are one to one and give their inverses.

So I believe something like T1(0) = 0, T2(1) = 1 and T3(2) = 2 will be the first three functions? How would you get the other three functions?

September 28th 2010, 06:41 AM

Ackbeet

No, you've described, essentially, only one of the functions, and you have 5 left. To define a function on V, you must describe where each element gets mapped to by T. You could describe your first function there as (0,1,2) -> (0,1,2). Another function would be (0,1,2) -> (0,2,1). Do you know what the name is of this mathematical idea? That will help you get all the desired functions.

September 28th 2010, 06:49 AM

HappyJoe

Also, the condition T(V) = V means that the function T is surjective. It is a theorem, which you probably may find in your text book, that a surjective map from a finite set to itself is automatically one to one.

September 28th 2010, 06:56 AM

HallsofIvy

And, indeed, that means that the 6 functions are the 6 permutations on{0, 1, 2}.